le will be equal in length
to two sides of the hexagon or two radii of the circle, as _EF_, and its
width will be twice the height of an equilateral triangle _mon_.

[Illustration: Fig. 206.]

To put the hexagon into perspective, draw base of quadrilateral _AD_,
divide it into four equal parts, and from each division draw lines to
point of sight. From _h_ drop perpendicular _ho_, and form equilateral
triangle _mno_. Take the height _ho_ and measure it twice along the base
from _A_ to 2. From 2 draw line to point of distance, or from 1 to
1/2 distance, and so find length of side _AB_ equal to A2. Draw _BC_,
and _EF_ through centre _o'_, and thus we have the six points through
which to draw the hexagon.

[Illustration: Fig. 207.]

CXV

A PAVEMENT COMPOSED OF HEXAGONAL TILES

In drawing pavements, except in the cases of square tiles, it is
necessary to make a plan of the required design, as in this figure
composed of hexagons. First set out the hexagon as at _A_, then draw
parallels 1 1, 2 2, &c., to mark the horizontal ends of the tiles
and the intermediate lines _oo_. Divide the base into the required
number of parts, each equal to one side of the hexagon, as 1, 2, 3, 4,
&c.; from these draw perpendiculars as shown in the figure, and also the
diagonals passing through their intersections. Then mark with a strong
line the outlines of the hexagonals, shading some of them; but the
figure explains itself.

It is easy to put all these parallels, perpendiculars, and diagonals
into perspective, and then to draw the hexagons.

First draw the hexagon on _AD_ as in the previous figure, dividing _AD_
into four, &c., set off right and left spaces equal to these fourths,
and from each division draw lines to point of sight. Produce sides _me_,
_nf_ till they touch the horizon in points _V_, _V'_; these will be the
two vanishing points for all the sides of the tiles that are receding
from us. From each division on base draw lines to each of these
vanishing points, then draw parallels through their intersections as
shown on the figure. Having all these guiding lines it will not be
difficult to draw as many hexagons as you please.

[Illustration: Fig. 208.]

Note that the vanishing points should be at equal distances from _S_,
also that the parallelogram in which each tile is contained is oblong,
and not square, as already pointed out.

We have also made use of the triangle _omn_ to ascertain the length and
w